Abstract

We first introduce the concept of manageable functions and then prove some new existence theorems related to approximate fixed point property for manageable functions and -admissible multivalued maps. As applications of our results, some new fixed point theorems which
generalize and improve Du's fixed point theorem, Berinde-Berinde's fixed point theorem, Mizoguchi-Takahashi's fixed point theorem, and Nadler's fixed point theorem and some well-known results in the literature are given.

1. Introduction and Preliminaries

In 1922, Banach established the most famous fundamental fixed point theorem (so-called the Banach contraction principle [1]) which has played an important role in various fields of applied mathematical analysis. It is known that the Banach contraction principle has been extended and generalized in many various different directions by several authors; see [2–40] and references therein. An interesting direction of research is the extension of the Banach contraction principle to multivalued maps, known as Nadler’s fixed point theorem [2], Mizoguchi-Takahashi’s fixed point theorem [3], and Berinde-Berinde’s fixed point theorem [5] and references therein.

Let us recall some basic notations, definitions, and well-known results needed in this paper. Throughout this paper, we denote by and the sets of positive integers and real numbers, respectively. Let be a metric space. For each and , let . Denote by the class of all nonempty subsets of , the family of all nonempty closed subsets of , and the family of all nonempty closed and bounded subsets of . A function defined by
is said to be the Hausdorff metric on induced by the metric on . A point in is a fixed point of a map , if (when is a single-valued map) or (when is a multivalued map). The set of fixed points of is denoted by . The map is said to have the approximate fixed point property [29–34] on provided . It is obvious that implies that has the approximate fixed point property, but the converse is not always true.

Definition 1 (see [6, 13]). A function is said to be an -function (or -function) if for all .

It is evident that if is a nondecreasing function or a nonincreasing function, then is a -function. So the set of -functions is a rich class.

Recently, Du [6] first proved the following characterizations of -functions which are quite useful for proving our main results.

Theorem 2 (see [6]). Let be a function. Then the following statements are equivalent.(a)is an -function.(b)For each , there exist and such that for all .(c)For each , there exist and such that for all .(d)For each , there exist and such that for all .(e)For each , there exist and such that for all .(f)For any nonincreasing sequence in , one has .(g) is a function of contractive factor [15]; that is, for any strictly decreasing sequence in , one has .

In 1989, Mizoguchi and Takahashi [3] proved a famous generalization of Nadler’s fixed point theorem which gives a partial answer of Problem 9 in Reich [4].

Theorem 3 (Mizoguchi and Takahashi [3]). Let be a complete metric space, let be an -function, and let be a multivalued map. Assume that
for all . Then .

In 2007, M. Berinde and V. Berinde [5] proved the following interesting fixed point theorem which generalized and extended Mizoguchi-Takahashi’s fixed point theorem.

Theorem 4 (M. Berinde and V. Berinde [5]). Let be a complete metric space, let be an -function, let be a multivalued map, and . Assume that
for all . Then .

In 2012, Du [6] established the following fixed point theorem which is an extension of Berinde-Berinde’s fixed point theorem and hence Mizoguchi-Takahashi’s fixed point theorem.

Theorem 5 (Du [6]). Let be a complete metric space, let be a multivalued map, let be a -function, and let be a function. Assume that
Then has a fixed point in .

The paper is organized as follows. In Section 2, we first introduce the concept of manageable function and give some examples of it. Section 3 is dedicated to the study of some new existence theorems related to approximate fixed point property for manageable functions and -admissible multivalued maps. As applications of our results, some new fixed point theorems which generalize and improve Du’s fixed point theorem, Berinde-Berinde’s fixed point theorem, Mizoguchi-Takahashi’s fixed point theorem, and Nadler’s fixed point theorem and some well-known results in the literature are given in Section 4. Consequently, some of our results in this paper are original in the literature, and we obtain many results in the literature as special cases.

2. Manageable Functions

In this paper, we first introduce the concept of manageable functions.

Definition 6. A function is called manageable if the following conditions hold: for all ;for any bounded sequence and any nonincreasing sequence , it holds that
We denote the set of all manageable functions by .

Here, we give simple examples of manageable function.

Example A. Let . Then defined by
is a manageable function.

Example B. Let be any function. Then the function defined by
is a manageable function. Indeed, let
Then for all , and
For any , we have
so holds. Let be a bounded sequence and let be a nonincreasing sequence. Then for some . Since is continuous, we get
which means that holds. Hence, .

Example C. Let be any function and let be an -function. Define by
Then is a manageable function. Indeed, one can verify easily that holds. Next, we verify that satisfies . Let be a bounded sequence and let be a nonincreasing sequence. Then for some . Since is an -function, by Theorem 2, there exist and such that for all . Since , there exists , such that
Hence, we have
which means that holds. So we prove .

The following result is quite obvious.

Proposition 7. Let be a function. If there exists such that for all , then .

Proof. Since for all , the conclusion (a) is a direct consequence of Proposition 7. Next, we prove the conclusion (b). Let be given. It is obvious that for all . Let be a bounded sequence and let be a nonincreasing sequence. For any , we have
Because each satisfies , we get
Hence, for each , the function is a manageable function.

3. New Existence Results for Manageable Functions and Approximate Fixed Point Property

Recall that a multivalued map is called(1)a Nadler’s type contraction (or a multivalued -contraction [3, 33]), if there exists a number such that
(2)a Mizoguchi-Takahashi’s type contraction [33], if there exists an -function such that
(3)a multivalued -almost contraction [28, 29, 33], if there exist two constants and such that
(4)a Berinde-Berinde’s type contraction [33] (or a generalized multivalued almost contraction [28, 29, 33]), if there exists an -function and such that
(5)a Du’s strong type contraction, if there exist an -function and a function such that
(6)a Du’s weak type contraction, if there exist an -function and a function such that

Definition 9 (see [36–39]). Let be a metric space and let be a multivalued map. One says that is -admissible, if there exists a function such that for each and with , one has for all .

The following existence theorem is one of the main results of this paper.

Theorem 10. Let be a metric space, let be an -admissible multivalued map, and . Let
If for all and there exist and such that , then the following statements hold.(a)There exists a Cauchy sequence in such that(i) for all ,(ii) for all ,(iii).(b); that is, has the approximate fixed point property on .

Proof. By our assumption, there exist and such that . If , then and
which implies . Let for all . Then is a Cauchy sequence in and
Clearly, for all . Hence, the conclusions (a) and (b) hold in this case. Assume or . If , then, following a similar argument as above, we can prove the conclusions (a) and (b) by taking a Cauchy sequence with and for all . Suppose . Thus . Define by
By , we know that
Since and for all , we have
Clearly, . So, by (29), we obtain
Let
Taking into account , , and the last inequality, we get . Since
there exists such that and
If , then the proof can be finished by a similar argument as above. Otherwise, we have . Since is -admissible, we obtain . By taking
then there exists with such that
By induction, if is known satisfying , , , , and
then, by taking
one can obtain with such that
Hence, by induction, we can establish sequences in satisfying, for each ,
By (30), we have
Hence, for each , by combining (40) and (41), we get
which means that the sequence is strictly decreasing in . So
By (41), we have
which means that is a bounded sequence. By , we have
Now, we claim . Suppose . Then, by (45) and taking lim sup in (42), we get
a contradiction. Hence we prove
To complete the proof of (a), it suffices to show that is a Cauchy sequence in . For each , let
Then for all . By (42), we obtain
From (45), we have , so there exist and , such that
For any , since for all and , taking into account (49) and (50) concludes that
Put , . For with , we have from the last inequality that
Since , . Hence
So is a Cauchy sequence in . Let for all . Then is the desired Cauchy sequence in (a).To see (b), since for each , we have
Combining (47) and (54) yields
The proof is completed.

Applying Theorem 10, we can establish the following new existence theorem related to approximate fixed point property for -admissible multivalued maps.

Theorem 11. Let be a metric space and let be an -admissible multivalued map. Suppose that there exists an -function such that
If there exist and such that , then the following statements hold.(a)There exists such that for all , where
(b)There exists a Cauchy sequence in such that(i) for all ,(ii) for all ,(iii).(c); that is, has the approximate fixed point property on .

Proof. Define by
By Example C, we know . By (56), we obtain for all . Therefore (a) is proved. It is obvious that the desired conclusions (b) and (c) follow from Theorem 10 immediately.

Corollary 12. Let be a metric space and let be an -admissible multivalued map. Assume that one of the following conditions holds.(L1)There exist an -function and a function such that
(L2)there exist an -function and such that
(L3)there exist two constants and such that
(L4)there exists an -function such that
(L5)there exists a number such that
If there exist and such that , then the following statements hold.(a)There exists such that for all , where
(b)There exists a Cauchy sequence in such that(i) for all ,(ii) for all ,(iii).(c); that is, has the approximate fixed point property on .

Proof. It suffices to verify the conclusion under (L1). Note first that, for each , for all . So, for each , by (L1), we obtain
which means (56) holds. Therefore, the conclusion follows from Theorem 11.

In Corollary 12, if we take by for all , then we obtain the following existence theorem.

Corollary 13. Let be a metric space and let be a multivalued map. Assume that one of the following conditions holds.(1) is a Du’s weak type contraction;(2) is a Du’s strong type contraction;(3) is a Berinde-Berinde’s type contraction;(4) is a multivalued -almost contraction;(5) is a Mizoguchi-Takahashi’s type contraction;(6) is a Nadler’s type contraction.Then the following statements hold.(a)There exists such that for all , where
(b)There exists a Cauchy sequence in such that(i) for all ,(ii).(c); that is, has the approximate fixed point property on .

4. Some Applications to Fixed Point Theory

Definition 14 (see [36–39]). Let be a metric space and let be a function. is said to have the property if any sequence in with for all and , we have for all .

Theorem 15. Let be a complete metric space and let be an -admissible multivalued map. Suppose that there exists an -function such that
If there exist and such that , and one of the following conditions is satisfied:(H1) is -continuous (i.e., implies as );(H2) is closed (i.e., ; the graph of is a closed subset of );(H3)the map defined by is l.s.c.;(H4)for any sequence in with , , , and , one has , then admits a fixed point in .

Proof. Applying Theorem 11, there exists a Cauchy sequence in such that
By the completeness of , there exists such that as .Now, we verify . If (H1) holds, since is -continuous on , for each , and as , we get
which implies . By the closeness of , we have . If (H2) holds, since is closed, for each , and as , we have . Suppose that (H3) holds. Since is convergent in , we have
Since
we obtain , and hence . Finally, assume (H4) holds. Then we obtain
Hence . Therefore, in any case, we prove . This completes the proof.

Theorem 16. Let be a complete metric space and let be an -admissible multivalued map. Suppose that there exist an -function and a function such that
If there exist and such that , and one of the following conditions is satisfied:(S1) is -continuous;(S2) is closed;(S3)the map defined by is l.s.c.;(S4)the function has the property , then admits a fixed point in .

Proof. It is obvious that (73) implies (67). If one of the conditions (S1), (S2), and (S3) is satisfied, then the desired conclusion follows from Theorem 15 immediately. Suppose that (S4) holds. We claim that (H4) as in Theorem 15 is satisfied. Let be in with , , , and . Since has the property , for all . So, it follows from (73) that
which implies . Hence (H4) holds. By Theorem 15, we also prove . The proof is completed.

Applying Theorem 16, we can give a short proof of Du’s fixed point theorem.

Corollary 17 (Du [[6]). Let be a complete metric space, let be a multivalued map, let be a -function, let and be a function. Assume that
Then .

Proof. Take by for all . Then (75) implies (73). Moreover, is an -admissible multivalued map and the function has the property . Therefore the conclusion follows from Theorem 16.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The first author was supported by grant no. NSC 102-2115-M-017-001 of the National Science Council of the Republic of China. The second author would like to express his sincere thanks to the Arak branch of Islamic Azad University for supporting this work.